Question: "Old Testament vs. New Testament - What are the differences?"
Answer: The Old Testament lays the foundation for the teachings and events found in the New Testament. The Bible is a progressive revelation. If you skip the first half of any good book and try to finish it, you will have a hard time understanding the characters, the plot, and the ending. In the same way, the New Testament is only completely understood when it is seen as a fulfillment of the events, characters, laws, sacrificial system, covenants, and promises of the Old Testament.
If we only had the New Testament, we would come to the gospels and not know why the Jews were looking for a Messiah (a Savior King). Without the Old Testament, we would not understand why this Messiah was coming (see Isaiah 53), and we would not have been able to identify Jesus of Nazareth as the Messiah through the many detailed prophecies that were given concerning Him, e.g., His birth place (Micah 5:2); His manner of death (Psalm 22, especially vv. 1,7-8, 14-18; Psalm 69:21, etc.), His resurrection (Psalm 16:10), and many more details of His ministry (Isaiah 52:19f.; 9:2, etc.).
Without the Old Testament, we would not understand the Jewish customs that are mentioned in passing in the New Testament. We would not understand the perversions that the Pharisees had made to God's law as they added their traditions to it. We would not understand why Jesus was so upset as He cleansed the temple courtyard. We would not understand that we can make use of the same wisdom that Christ used in His many replies to His adversaries (both human and demonic).
In a similar fashion, the New Testament Gospels and Acts of the Apostles record many of the fulfillments of prophecies that were recorded hundreds of years earlier in the Old Testament. Many of these relate to the first coming of the Messiah. In the circumstances of Jesus' birth, life, miracles, death, and resurrection as found in the Gospels, we find the fulfillment of the Old Testament prophecies that relate to the Messiah's first coming. It is these details that validate Jesus' claim to be the promised Christ. And even the prophecies in the New Testament (many of which are in the book of Revelation) are built upon earlier prophecies found in Old Testament books. These New Testament prophecies relate to events surrounding the second coming of Christ. Roughly two out of three verses in Revelation are based on Old Testament verses.
Both the Old and New Testaments contain numerous lessons for us through the lives of its many fallible characters who possessed the same nature as we do today. By observing their lives we can be encouraged to trust God no matter what (Daniel 3) and to not compromise in the little things (Daniel 1) so that we will be faithful later in the big things (Daniel 6). We can learn that it is best to confess sin early and sincerely instead of shifting blame (1 Samuel 15). We can learn not to play with sin, it will find us out and its bite is deadly (See Judges 13-16).
We can learn that we need to trust and obey God if we expect to experience His "promised land" living in this life and His paradise in the next (Numbers 13). We learn that if we contemplate sin, we are only setting ourselves up for committing it (Genesis 3; Joshua 6-7). We learn that our sin has consequences not only for ourselves but for our loved ones around us and conversely that our good behavior has reward for us and those around us as well (Genesis 3; Exodus 20:5-6). In the New Testament, we have the example of Peter to learn from—that we dare not trust our own strength or we WILL fail (Matthew 26:33-41). In the words of the thief on the cross, we see that it is through simple, sincere faith that we are saved from our sin (Luke 23:39-43). We also see what a vital New Testament church should look like (Acts 2:41-47; 13:1-3, etc.).
Also, because the revelation in Scripture is progressive, the New Testament brings into focus teachings that were only alluded to in the Old Testament. The book of Hebrews describes how Jesus is the true High Priest and His one sacrifice replaces all of the sacrifices that were mere portrayals of that sacrifice. The Old Testament gives the Law which has two parts: the commandments and the blessing/curse that comes from obedience or disobedience to those commands. The New Testament clarifies that God gave those commandments to show men their need of salvation and were never intended to be a means of salvation (Romans 3:19).
The Old Testament describes the sacrificial system God gave the Israelites to temporarily cover their sins. The New Testament clarifies that this system was an allusion to the sacrifice of Christ through whom alone salvation is found (Acts 4:12; Hebrews 10:4-10). The Old Testament saw paradise lost; the New Testament shows how paradise was regained for mankind through the second Adam (Christ) and how it will one day be restored. The Old Testament declares that man was separated from God through sin (Genesis 3), and the New Testament declares that man can now be restored in his relationship to God (Romans 3-6). The Old Testament predicted the Messiah's life. The Gospels primarily record Jesus' life, and the Epistles interpret His life and how we are to respond to all He has done and will do.
Again, while the New Testament is the "clearer" picture, the Old Testament is nonetheless important. Beside laying the foundation for the New Testament, without the Old Testament we would not have a basis for standing against the error of the politically correct perversions of our society in which evolution is seen to be the creator of all of the species over millions of years (instead of their being the result of special creation by God in a literal six days). We would buy the lie that marriages and the family unit are an evolving structure that should continue to change as society changes (instead of being seen as a design by God for the purpose of raising up godly children and for the protection of those who would otherwise be used and abused—most often women and children).
Likewise, without the Old Testament we would not understand the promises God will yet fulfill to the Jewish nation. As a result, we would not properly see that the Tribulation period is a seven-year period in which He will specifically be working with the Jewish nation who rejected His first coming but who will receive Him at His second coming. We would not understand how Christ's future 1,000-year reign fits in with His promises to the Jews, nor how we as Gentiles will fit in. Nor would we see how the end of the Bible ties up the loose ends that were unraveled in the beginning of the Bible, restoring the paradise that God originally created this world to be in which we would enjoy close companionship with Him on a personal basis as in the Garden of Eden.
In summary, the Old Testament lays the foundation for, and was meant to prepare the Israelites for, the coming of the Messiah who would sacrifice Himself for their sins (and for the sins of the world as well). The New Testament shares the life of Jesus Christ and then looks back on what He did and how we are to respond to His gift of eternal life and live our lives in gratitude for all He has done for us (Romans 12). Both testaments reveal the same holy, merciful, and righteous God who must condemn sin but who desires to bring to Himself a fallen human race of sinners through the forgiveness only possible through Christ's atoning sacrifice as payment for sin. In both testaments God reveals Himself to us and how we are to come to Him through Jesus Christ. And in both testaments we find all we need for eternal life and godly living (2 Timothy 3:15-17).